Finite Group Databasebeta

← #13155840

Subgroups — #13155840

Order 1728.

860 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89041C211yesmaximal, characteristic, complemented
2#11723522C211yesmaximal, characteristic, complemented
3C2×C3×#3716C211yesmaximal, characteristic, complemented
4C2×C3×#3714C211yesmaximal, characteristic, complemented
5#11722040C211yesmaximal, characteristic, complemented
6#11722034C211yesmaximal, characteristic, complemented
7C3×#63557C211yesmaximal, characteristic, complemented
8#10591520—31nomaximal
9#10590962—31nomaximal
10#10590823—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic
15C2×#6295—21no—
16C2×#6294—21no—
17#89041—22no—
18#89041—22no—
19C2×#6290—21no—
20C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C2×C32×C3⋊C8C2211yescharacteristic
23(C2×D3)⋊(C3⋊Dic3)—21no—
24C3×#3716—22no—
25C3×#3714—22no—
26C2×#3799—31no—
27C2×#3796—31no—
28C2×#3795—31no—
29#64147—31no—
30C2×C3×#214—31no—
31C2×C3×#213—31no—
32C2×C3×D24—31no—
33C2×C3×#204—31no—
34#63908—31no—
35#63908—31no—
36#63908—61no—
37C2×#3716D311yescharacteristic, complemented
38C2×#3714D311yescharacteristic, complemented
39#63648—31no—
40#63643—31no—
41C3×(C2×D3)⋊C8—31no—
42C3×(C2×D3)⋊C8—31no—
43#63566—31no—
44#63566—31no—
45#63566—61no—
46#63563—31no—
47#63563—31no—
48#63563—61no—
49#63557D311yescharacteristic, complemented
50C22×C3×C3⋊D3—21no—
51C22×C3×C3⋊D3D411yescharacteristic
52C22×C32×D3—21no—
53C2×C33×C4C2311yescharacteristic
54C2×C3⋊(C3⋊Dic3)—21no—
55C2×C3×C3⋊Dic3—21no—
56C2×C3×C3⋊Dic3D411yescharacteristic
57C3×C4⋊(C3⋊D3)—21no—
58C3×C4⋊(C3⋊D3)D412yescomplemented
59C3×C4×C3⋊D3—21no—
60C3×C4×C3⋊D3—21no—
61C3×#634—21no—
62C3×#634D412yescomplemented
63C32×D12—21no—
64C32×D12—21no—
65#6295—24no—
66#6294—24no—
67#6290—42no—
68C32×C3⋊C8—21no—
69C32×C3⋊C8—21no—
70#5172—91no—
71#5158—91no—
72#4734—91no—
73C2×C32×D4—31no—
74C2×#638—61no—
75C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
76C2×C4×C3⋊D3C2×D311yescharacteristic
77C2×#634C2×D311yescharacteristic
78C2×C3×D12—21no—
79C2×C3×D12—31no—
80C2×C3×D12—31no—
81C2×C3×D12—61no—
82C2×C3×D12C2×D311yescharacteristic
83C2×C3×D12C2×D311yescharacteristic
84C2×C3×C4×D3—31no—
85C2×C3×C4×D3—31no—
86C2×C3×C4×D3—61no—
87C2×C3×Dic6—31no—
88C2×C3×Dic6—31no—
89C2×C3×Dic6—61no—
90C2×#626—61no—
91C2×#626—61no—
92C2×#625—61no—
93C2×#625—61no—
94C2×#625—61no—
95C2×#625—62no—
96C2×#624—61no—
97C2×D3×Dic3—61no—
98C2×D3×Dic3—61no—
99#3799—62no—
100#3799—62no—
101#3796—62no—
102#3796—62no—
103#3795—62no—
104#3795—62no—
105C2×C32×C8—31no—
106C2×C32×C8—31no—
107#3757—61no—
108C4⋊(C3⋊Dic3)—31no—
109C4⋊(C3⋊Dic3)—31no—
110C4⋊(C3⋊Dic3)—31no—
111C4⋊(C3⋊Dic3)—61no—
112C4⋊(C3⋊Dic3)—61no—
113C4⋊(C3⋊Dic3)—61no—
114C4⋊(C3⋊Dic3)—121no—
115C3×#214—62no—
116C3×#213—62no—
117C2×C3×C3⋊C8—21no—
118C2×C3×C3⋊C8—21no—
119C2×C3×C3⋊C8C2×D311yescharacteristic
120C2×C3×C3⋊C8C2×D311yescharacteristic
121C2×C3×C3⋊C8C2×D311yescharacteristic
122C2×C3×C3⋊C8C2×D311yescharacteristic
123C3×D24—62no—
124C3×#204—62no—
125(C2×D3)⋊Dic3—61no—
126(C2×D3)⋊Dic3—61no—
127(C2×D3)⋊Dic3—62no—
128#3716—22no—
129#3714—22no—
130C22×C33C2×D411yescharacteristic
131C2×C3×C3⋊D3—22no—
132C2×C3×C3⋊D3—21no—
133C2×C3×C3⋊D3—21no—
134C2×C3×C3⋊D3—21no—
135C2×C3×C3⋊D3—41no—
136C2×C3×C3⋊D3—41no—
137C2×C32×D3—41no—
138C2×C32×D3—41no—
139C2×C32×D3—41no—
140C33×C4C2×D411yescharacteristic
141C33×C4C2×D411yescharacteristic
142C3⋊(C3⋊Dic3)—41no—
143C3×C3⋊Dic3—21no—
144C3×C3⋊Dic3—22no—
145C2×#239—91no—
146C2×#237—91no—
147C2×#235—91no—
148C2×C3×SD16—91no—
149C2×C3×D8—91no—
150#938—91no—
151#938—91no—
152#938—181no—
153C2×#214—31no—
154C2×#213—31no—
155#930—91no—
156C2×D24—31no—
157C2×#204—31no—
158C3×C22⋊C8—91no—
159#835—91no—
160#832—91no—
161#821—91no—
162#821—91no—
163#821—181no—
164(C2×D3)⋊C8—31no—
165(C2×D3)⋊C8—31no—
166#816—91no—
167#816—91no—
168#816—181no—
169C23×C32—61no—
170C22×C3⋊D3—21no—
171C22×C3⋊D3D1211yescharacteristic
172C22×C3×D3—22no—
173C22×C3×D3—21no—
174C22×C3×D3—21no—
175C22×C3×D3—31no—
176C22×C3×D3—31no—
177C22×C3×D3—61no—
178C22×C3×D3—61no—
179C22×C3×D3—61no—
180C22×C3×D3—121no—
181C32×D4—61no—
182C32×D4—61no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4—41no—
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D311yescharacteristic
190#638—122no—
191C2×C3⋊Dic3—21no—
192C2×C3⋊Dic3—61no—
193C2×C3⋊Dic3—62no—
194C2×C3⋊Dic3—61no—
195C2×C3⋊Dic3—61no—
196C2×C3⋊Dic3—121no—
197C2×C3⋊Dic3—121no—
198C2×C3⋊Dic3—122no—
199C2×C3⋊Dic3D1211yescharacteristic
200C4⋊(C3⋊D3)—21no—
201C4⋊(C3⋊D3)#6712yescomplemented
202C4×C3⋊D3—21no—
203C4×C3⋊D3—21no—
204#634—21no—
205#634#6712yescomplemented
206C2×C3×Dic3—31no—
207C2×C3×Dic3—31no—
208C2×C3×Dic3—61no—
209C2×C3×Dic3—61no—
210C2×C3×Dic3—61no—
211C2×C3×Dic3—121no—
212C3×D12—21no—
213C3×D12—21no—
214C3×D12—21no—
215C3×D12—21no—
216C3×D12—32no—
217C3×D12—32no—
218C3×D12—41no—
219C3×D12—41no—
220C3×D12—62no—
221C3×D12—61no—
222C3×D12—61no—
223C3×D12—121no—
224C3×C4×D3—61no—
225C3×C4×D3—61no—
226C3×C4×D3—61no—
227C3×C4×D3—61no—
228C3×C4×D3—121no—
229C3×C4×D3—121no—
230C3×Dic6—32no—
231C3×Dic6—32no—
232C3×Dic6—62no—
233C3×Dic6—61no—
234C3×Dic6—61no—
235C3×Dic6—121no—
236#626—64no—
237#626—64no—
238#625—64no—
239#625—124no—
240#625—122no—
241#625—122no—
242#624—64no—
243D3×Dic3—64no—
244D3×Dic3—64no—
245C32×C8—61no—
246C32×C8—61no—
247C3×C3⋊C8—21no—
248C3×C3⋊C8—21no—
249C3×C3⋊C8—21no—
250C3×C3⋊C8—21no—
251C3×C3⋊C8—41no—
252C3×C3⋊C8—41no—
253#449—271noSylow
254C2×C33#12011yescharacteristic
255C2×C33#13511yescharacteristic
256C2×C33AGL(1,ℤ/8)11yescharacteristic
257C3×C3⋊D3—41no—
258C3×C3⋊D3—42no—
259C32×D3—81no—
260C2×C3×Q8—91no—
261C2×C3×D4—31no—
262C2×C3×D4—31no—
263C2×C3×D4—61no—
264C2×C3×D4—91no—
265C22×C3×C4—91no—
266C2×#67—181no—
267C2×#67—181no—
268C2×#67—181no—
269C2×#67—181no—
270C2×#67—181no—
271C2×#67—182no—
272C22×Dic3—181no—
273#239—182no—
274#239—182no—
275#237—182no—
276#237—182no—
277#235—182no—
278#235—182no—
279C2×D12—31no—
280C2×D12—31no—
281C2×D12—61no—
282C2×D12—181no—
283C2×D12D3211yescharacteristic
284C2×C4×D3—31no—
285C2×C4×D3—31no—
286C2×C4×D3—61no—
287C2×C4×D3—181no—
288C2×C4×D3—181no—
289C2×Dic6—31no—
290C2×Dic6—31no—
291C2×Dic6—61no—
292C3×SD16—182no—
293C3×D8—182no—
294C2×C3×C8—31no—
295C2×C3×C8—31no—
296C2×C3×C8—31no—
297C2×C3×C8—31no—
298C2×C3×C8—61no—
299C2×C3×C8—61no—
300#217—181no—
301#217—181no—
302#217—182no—
303#214—62no—
304#213—62no—
305(C2×D3)⋊C4—181no—
306C4⋊Dic3—91no—
307C4⋊Dic3—91no—
308C4⋊Dic3—91no—
309C4⋊Dic3—181no—
310C4⋊Dic3—181no—
311C4⋊Dic3—181no—
312C4⋊Dic3—361no—
313C2×C3⋊C8D3211yescharacteristic
314C2×C3⋊C8D3211yescharacteristic
315D24—62no—
316#204—62no—
317C22×C32—21no—
318C22×C32—21no—
319C22×C32—21no—
320C22×C32—41no—
321C22×C32—121no—
322C22×C32—121no—
323C22×C32—121no—
324C22×C32C2×D1211yescharacteristic
325C22×C32D3×D411yescharacteristic
326C22×C32D3×D411yescharacteristic
327C2×C3⋊D3—21no—
328C2×C3⋊D3—21no—
329C2×C3⋊D3—21no—
330C2×C3⋊D3—22no—
331C2×C3⋊D3—41no—
332C2×C3⋊D3—41no—
333C2×C3×D3—42no—
334C2×C3×D3—42no—
335C2×C3×D3—42no—
336C2×C3×D3—41no—
337C2×C3×D3—41no—
338C2×C3×D3—41no—
339C2×C3×D3—41no—
340C2×C3×D3—41no—
341C2×C3×D3—41no—
342C2×C3×D3—62no—
343C2×C3×D3—62no—
344C2×C3×D3—61no—
345C2×C3×D3—61no—
346C2×C3×D3—61no—
347C2×C3×D3—61no—
348C2×C3×D3—62no—
349C2×C3×D3—61no—
350C2×C3×D3—61no—
351C2×C3×D3—121no—
352C2×C3×D3—121no—
353C2×C3×D3—122no—
354C2×C3×D3—121no—
355C2×C3×D3—121no—
356C2×C3×D3—121no—
357C2×C3×D3—121no—
358C2×C3×D3—241no—
359C2×C3×D3—241no—
360C32×C4—21no—
361C32×C4—21no—
362C32×C4—21no—
363C32×C4—21no—
364C32×C4—21no—
365C32×C4—21no—
366C32×C4—41no—
367C32×C4—41no—
368C32×C4D3×D411yescharacteristic
369C32×C4D3×D411yescharacteristic
370C32×C4D3×D411yescharacteristic
371C32×C4D3×D411yescharacteristic
372C32×C4C2×#6711yescharacteristic
373C32×C4C2×#6711yescharacteristic
374C3⋊Dic3—21no—
375C3⋊Dic3—22no—
376C3⋊Dic3—121no—
377C3⋊Dic3—122no—
378C3⋊Dic3—121no—
379C3⋊Dic3—121no—
380C3⋊Dic3—241no—
381C3⋊Dic3—241no—
382C3⋊Dic3—242no—
383C3×Dic3—62no—
384C3×Dic3—61no—
385C3×Dic3—61no—
386C3×Dic3—62no—
387C3×Dic3—122no—
388C3×Dic3—121no—
389C2×#41—271no—
390C2×SD16—91no—
391C2×D8—91no—
392#121—271no—
393#103—271no—
394#102—271no—
395C22⋊C8—91no—
396C33#44911yescharacteristic, complemented, Sylow
397C23×C3—62no—
398C23×C3—61no—
399C23×C3—61no—
400C23×C3—91no—
401C23×C3—181no—
402C22×D3—21no—
403C22×D3—31no—
404C22×D3—31no—
405C22×D3—61no—
406C22×D3—61no—
407C22×D3—61no—
408C22×D3—121no—
409C3×Q8—92no—
410C3×Q8—181no—
411C3×D4—61no—
412C3×D4—61no—
413C3×D4—61no—
414C3×D4—61no—
415C3×D4—92no—
416C3×D4—121no—
417C3×D4—121no—
418C3×D4—181no—
419C2×C3×C4—21no—
420C2×C3×C4—21no—
421C2×C3×C4—21no—
422C2×C3×C4—41no—
423C2×C3×C4—91no—
424C2×C3×C4—181no—
425C2×C3×C4—181no—
426C2×C3×C4—181no—
427C2×C3×C4C2×D3211yescharacteristic
428C2×C3×C4C2×D3211yescharacteristic
429C2×C3×C4C2×D3211yescharacteristic
430#67—184no—
431#67—184no—
432#67—362no—
433#67—364no—
434#67—362no—
435#67—362no—
436C2×Dic3—31no—
437C2×Dic3—31no—
438C2×Dic3—61no—
439C2×Dic3—61no—
440C2×Dic3—61no—
441C2×Dic3—121no—
442C2×Dic3—184no—
443C2×Dic3—182no—
444C2×Dic3—181no—
445C2×Dic3—181no—
446C2×Dic3—181no—
447C2×Dic3—361no—
448C2×Dic3—361no—
449C2×Dic3—362no—
450D12—21no—
451D12—21no—
452D12—32no—
453D12—32no—
454D12—62no—
455D12—61no—
456D12—61no—
457D12—121no—
458D12—184no—
459C4×D3—61no—
460C4×D3—61no—
461C4×D3—61no—
462C4×D3—61no—
463C4×D3—121no—
464C4×D3—121no—
465C4×D3—184no—
466C4×D3—184no—
467Dic6—32no—
468Dic6—32no—
469Dic6—62no—
470Dic6—61no—
471Dic6—61no—
472Dic6—121no—
473C3×C8—61no—
474C3×C8—61no—
475C3×C8—61no—
476C3×C8—61no—
477C3×C8—121no—
478C3×C8—121no—
479C3⋊C8—21no—
480C3⋊C8—21no—
481C2×C32—21no—
482C2×C32—21no—
483C2×C32—21no—
484C2×C32—21no—
485C2×C32—21no—
486C2×C32—21no—
487C2×C32—21no—
488C2×C32—21no—
489C2×C32—21no—
490C2×C32—41no—
491C2×C32—41no—
492C2×C32—41no—
493C2×C32—241no—
494C2×C32#88211yescharacteristic
495C2×C32#90411yescharacteristic
496C2×C32#90811yescharacteristic
497C2×C32#91111yescharacteristic
498C2×C32#91211yescharacteristic
499C2×C32#91411yescharacteristic
500C2×C32#91611yescharacteristic
501C2×C32#93711yescharacteristic
502C2×C32#93711yescharacteristic
503C3⋊D3—41no—
504C3⋊D3—42no—
505C3×D3—82no—
506C3×D3—81no—
507C3×D3—81no—
508C3×D3—122no—
509C3×D3—121no—
510C3×D3—121no—
511C3×D3—122no—
512C3×D3—122no—
513C3×D3—242no—
514#41—542no—
515#41—542no—
516C2×Q8—91no—
517C2×D4—31no—
518C2×D4—91no—
519C2×D4—541no—
520C2×D4—541no—
521C22×C4—91no—
522C22×C4—541no—
523SD16—182no—
524D8—182no—
525C2×C8—31no—
526C2×C8—31no—
527C4⋊C4—271no—
528C22⋊C4—541no—
529C22×C3—21no—
530C22×C3—21no—
531C22×C3—21no—
532C22×C3—41no—
533C22×C3—122no—
534C22×C3—122no—
535C22×C3—122no—
536C22×C3—121no—
537C22×C3—121no—
538C22×C3—121no—
539C22×C3—121no—
540C22×C3—121no—
541C22×C3—121no—
542C22×C3—182no—
543C22×C3—181no—
544C22×C3—181no—
545C22×C3—181no—
546C22×C3—361no—
547C22×C3—361no—
548C22×C3D3×D1211yescharacteristic
549C22×C3D3×D1211yescharacteristic
550C22×C3#381111yescharacteristic
551C2×D3—41no—
552C2×D3—41no—
553C2×D3—41no—
554C2×D3—62no—
555C2×D3—61no—
556C2×D3—61no—
557C2×D3—61no—
558C2×D3—61no—
559C2×D3—61no—
560C2×D3—61no—
561C2×D3—62no—
562C2×D3—62no—
563C2×D3—121no—
564C2×D3—121no—
565C2×D3—122no—
566C2×D3—121no—
567C2×D3—121no—
568C2×D3—121no—
569C2×D3—121no—
570C2×D3—241no—
571C2×D3—241no—
572C3×C4—21no—
573C3×C4—21no—
574C3×C4—21no—
575C3×C4—21no—
576C3×C4—21no—
577C3×C4—21no—
578C3×C4—41no—
579C3×C4—41no—
580C3×C4—182no—
581C3×C4—181no—
582C3×C4#380211yescharacteristic
583C3×C4#380211yescharacteristic
584C3×C4D3×#6711yescharacteristic
585C3×C4D3×#6711yescharacteristic
586C3×C4D3×#6711yescharacteristic
587C3×C4D3×#6711yescharacteristic
588Dic3—62no—
589Dic3—61no—
590Dic3—61no—
591Dic3—62no—
592Dic3—122no—
593Dic3—121no—
594Dic3—362no—
595Dic3—361no—
596Dic3—361no—
597Dic3—361no—
598Dic3—721no—
599Dic3—721no—
600Dic3—722no—
601C32—21no—
602C32—21no—
603C32—21no—
604C32—41no—
605C32#473411yescharacteristic, complemented
606C32#515811yescharacteristic, complemented
607C32#517211yescharacteristic, complemented
608C23—61no—
609C23—91no—
610C23—181no—
611Q8—92no—
612Q8—181no—
613D4—61no—
614D4—61no—
615D4—92no—
616D4—181no—
617D4—544no—
618D4—1082no—
619C2×C4—91no—
620C2×C4—181no—
621C2×C4—181no—
622C2×C4—181no—
623C2×C4—544no—
624C2×C4—541no—
625C2×C4D3311yescharacteristic
626C8—61no—
627C8—61no—
628C2×C3—21no—
629C2×C3—21no—
630C2×C3—21no—
631C2×C3—21no—
632C2×C3—21no—
633C2×C3—21no—
634C2×C3—21no—
635C2×C3—21no—
636C2×C3—21no—
637C2×C3—41no—
638C2×C3—41no—
639C2×C3—41no—
640C2×C3—242no—
641C2×C3—241no—
642C2×C3—241no—
643C2×C3—361no—
644C2×C3—362no—
645C2×C3#6379411yescharacteristic
646C2×C3#6379511yescharacteristic
647C2×C3#6380611yescharacteristic
648C2×C3#6380711yescharacteristic
649C2×C3#6392211yescharacteristic
650C2×C3#6392211yescharacteristic
651C2×C3#6393611yescharacteristic
652C2×C3#6393711yescharacteristic
653C2×C3#6398311yescharacteristic
654D3—81no—
655D3—122no—
656D3—122no—
657D3—121no—
658D3—121no—
659D3—122no—
660D3—242no—
661C22—121no—
662C22—121no—
663C22—121no—
664C22—181no—
665C22—181no—
666C22—181no—
667C22—182no—
668C22—361no—
669C22—361no—
670C22#8895511yescharacteristic
671C4—181no—
672C4—182no—
673C4—1081no—
674C4#8895311yescharacteristic
675C4#8895311yescharacteristic
676C3—21no—
677C3—21no—
678C3—21no—
679C3—41no—
680C3#1059082311yescharacteristic, complemented
681C3#1059096211yescharacteristic, complemented
682C3#1059152011yescharacteristic, complemented
683C2—241no—
684C2—361no—
685C2—362no—
686C2#1172286411yescharacteristic
687C2#1172287911yescharacteristic
688C2#1172295111yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
hypocenterC33#449
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#89041C2semidirect
2. #11723522C2semidirect
3. C2×C3×#3716C2semidirect
4. C2×C3×#3714C2semidirect
5. #11722040C2semidirect
6. #11722034C2semidirect
7. C3×#63557C2semidirect
37. C2×#3716D3semidirect
38. C2×#3714D3semidirect
49. #63557D3semidirect
58. C3×C4⋊(C3⋊D3)D4semidirect
62. C3×#634D4semidirect
201. C4⋊(C3⋊D3)#67semidirect
205. #634#67semidirect
396. C33#449semidirect
605. C32#4734semidirect
606. C32#5158semidirect
607. C32#5172semidirect
680. C3#10590823semidirect
681. C3#10590962semidirect
682. C3#10591520semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#890412
#117235222
C2×C3×#37162
C2×C3×#37142
#117220402
#117220342
C3×#635572
#105915203
#105909623
#105908233